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# initial package installation
Pkg.add("Convex")
Pkg.add("SCS")
Pkg.add("Gadfly")
Pkg.add("Interact")
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# Make the Convex.jl module available
using Convex
using SCS # first order splitting conic solver [O'Donoghue et al., 2014]
set_default_solver(SCSSolver(verbose=0)) # could also use Gurobi, Mosek, CPLEX, ...
# Generate random problem data
m = 50; n = 100
A = randn(m, n)
x♮ = sprand(n, 1, .5) # true (sparse nonnegative) parameter vector
noise = .1*randn(m) # gaussian noise
b = A*x♮ + noise # noisy linear observations
# Create a (column vector) variable of size n.
x = Variable(n)
# nonnegative elastic net with regularization
λ = 1
μ = 1
problem = minimize(norm(A * x - b)^2 + λ*norm(x)^2 + μ*norm(x, 1),
x >= 0)
# Solve the problem by calling solve!
solve!(problem)
println("problem status is ", problem.status) # :Optimal, :Infeasible, :Unbounded etc.
println("optimal value is ", problem.optval)
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using Gadfly, Interact
@manipulate for λ=0:.1:5, mu=0:.1:5
problem = minimize(norm(A * x - b)^2 + λ*norm(x)^2 + μ*norm(x, 1),
x >= 0)
solve!(problem)
plot(x=x.value, Geom.histogram(minbincount = 20),
Scale.x_continuous(minvalue=0, maxvalue=3.5))#, Scale.y_continuous(minvalue=0, maxvalue=6))
end
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# Scalar variable
x = Variable()
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# (Column) vector variable
y = Variable(4)
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# Matrix variable
Z = Variable(4, 4)
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Convex.jl allows you to use a wide variety of functions on variables and on expressions to form new expressions.
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x + 2x
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e = y[1] + logdet(Z) + sqrt(x) + minimum(y)
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e.children[2]
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x <= 0
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x^2 <= sum(y)
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M = Z
for i = 1:length(y)
M += rand(size(Z))*y[i]
end
M ⪰ 0
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x = Variable()
y = Variable(4)
objective = 2*x + 1 - sqrt(sum(y))
constraint = x >= maximum(y)
p = minimize(objective, constraint)
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# solve the problem
solve!(p)
p.status
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x.value
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# can evaluate expressions directly
evaluate(objective)
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call a MathProgBase solver suited for your problem class
to solve problem using a different solver, just import the solver package and pass the solver to the solve! method: eg
using Mosek
solve!(p, MosekSolver())
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# Generate random problem data
m = 50; n = 100
A = randn(m, n)
x♮ = sprand(n, 1, .5) # true (sparse nonnegative) parameter vector
noise = .1*randn(m) # gaussian noise
b = A*x♮ + noise # noisy linear observations
# Create a (column vector) variable of size n.
x = Variable(n)
# nonnegative elastic net with regularization
λ = 1
μ = 1
problem = minimize(norm(A * x - b)^2 + λ*norm(x)^2 + μ*norm(x, 1),
x >= 0)
@time solve!(problem)
λ = 1.5
@time solve!(problem, warmstart = true)
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# affine
x = Variable(4)
y = Variable (2)
sum(x) + y[2]
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2*maximum(x) + 4*sum(y) - sqrt(y[1] + x[1]) - 7 * minimum(x[2:4])
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# not dcp compliant
log(x) + x^2
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# $f$ is convex increasing and $g$ is convex
square(pos(x))
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# $f$ is convex decreasing and $g$ is concave
invpos(sqrt(x))
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# $f$ is concave increasing and $g$ is concave
sqrt(sqrt(x))
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